NEW METHOD FOR THE NUMERICAL SOLUTION OF THE FREDHOLM LINEAR INTEGRAL EQUATION ON A LARGE INTERVAL Текст научной статьи по специальности «Математика»

Том 25, № 132

(g Lemita S., Guebbai H., Sedka I., Aissaoui M.Z., 2020 DOI 10.20310/2686-9667-2020-25-132-387-400 UDC 519.642.4

2020

New method for the numerical solution of the Fredholm linear integral equation on a large interval

Samir LEMITA1 , Hamza GUEBBAI2 , Ilyes SEDKA2 , Mohamed Zine AISSAOUI2

1 Higher Normal School of Ouargla B.P. 398, Ennacer St., Ouargla 30000, Algeria

2 University May 8, 1945 - Guelma B.P. 401, Guelma 24000, Algeria

Новый метод численного решения линейного интегрального уравнения Фредгольма на большом интервале

Самир ЛЕМИТА1 , Хамза ГЕББАЙ2 , Ильес СЕДКА2 , Мохамед Зин АИССАУИ2

1 Высшая нормальная школа Уаргла 30000, Алжир, Уаргла, ул. Еннасер, П.Я. 398

2 Университет 8 мая 1945 г. - Гельма

24000, Алжир, Гельма, П.Я. 401

Abstract. The traditional numerical process to tackle a linear Fredholm integral equation on a large interval is divided into two parts, the first is discretization, and the second is the use of the iterative scheme to approach the solutions of the huge algebraic system. In this paper we propose a new method based on constructing a generalization of the iterative scheme, which is adapted to the system of linear bounded operators. Then we don't discretize the whole system, but only the diagonal part of the system. This system is built by transforming our integral equation. As discretization we consider the product integration method and the Gauss-Seidel iterative method as iterative scheme. We also prove the convergence of this new method. The numerical tests developed show its effectiveness.

Keywords: Fredholm equation of the second kind; weakly singular kernel; large integration interval; Gauss-Seidel method; bounded operators matrix; product integration method

For citation: Lemita S., Guebbai H., Sedka I., Aissaoui M.Z. Novyy metod chislennogo reshe-niya lineynogo integral'nogo uravneniya Fredgol'ma na bol'shom intervale [New method for the numerical solution of the Fredholm linear integral equation on a large interval]. Vestnik rossiyskikh universitetov. Matematika - Russian Universities Reports. Mathematics, 2020, vol. 25, no. 132, pp. 387-400. DOI 10.20310/2686-9667-2020-25-132-387-400.

Аннотация. Традиционное численное решение линейного интегрального уравнения Фредгольма на большом интервале делится на два этапа: первый — дискретизация, второй — использование итерационной схемы для приближения к решению алгебраической системы большой размерности (полученной на первом этапе). В этой статье мы предлагаем новый метод, основанный на построении обобщения итерационной схемы, которая адаптирована к системе линейных ограниченных операторов, при этом мы не дискретизируем всю систему, а только ее диагональную часть. Рассматриваемая система строится путем преобразования исходного интегрального уравнения. В качестве дискретизации мы рассматриваем метод интегрирования произведения, а в качестве итерационной схемы — итерационный метод Гаусса-Зайделя. Мы также анализируем сходимость этого нового метода. Численные тесты показывают его эффективность.

Ключевые слова: уравнение Фредгольма второго рода; слабо сингулярное ядро; большой интервал интегрирования; метод Гаусса-Зейделя; матрица ограниченных операторов; метод интегрирования произведений

Для цитирования: Лемита С., Геббай Х., Седка И., Аиссауи М.З. Новый метод численного решения линейного интегрального уравнения Фредгольма на большом интервале // Вестник российских университетов. Математика. 2020. Т. 25. № 132. С. 387-400. DOI 10.20310/2686-9667-2020-25-132-387-400. (In Engl., Abstr. in Russian)

Introduction

Numerical approximation of linear Fredholm integral equations leads to linear algebraic systems. The size of the matrices obtained in the linear systems depends on the order of convergence. So, we have to solve a huge system to get a small error. However, this system can not be solved directly, so we use the Gauss-Seidel iterative scheme to approach its solutions. For this iterative method, many variants have been developed [1-6].

In this paper we propose a new method. First we make our integral equation into a system of the following form:

' A«i = Tnui + T12M2 + ... + tin«N + /1, A«2 = T21W1 + T22W2 + ... + T2 n «N + /2,

a«N = TN i«i + TN 2«2 + ... + tNN «N + /N ,

where {Tj}1<ij<N is a family of bounded operators. Next we construct a generalization of the Gauss-Seidel method adapted to (0.1), after that we discretize only the diagonal part of this system to approach a solution of the initial equation.

In [7] and [8], a generalization of Jacobi's method adapted to the same system (0.1) has been constructed in order to approach a regular and a weakly singular Frehdolm integral equation, respectively. The numerical study of those papers presents very good results. In this paper, in a similar way, we construct a generalization of Gauss-Seidel method to approach a linear Fredholm integral equation of the second kind with weakly singular kernel defined on a large interval.

Let X = C ([0, t]) be the Banach space of continuous functions equipped with the norm

V^I IIXI X = max |x(t)| ,

11 0<t<r 1 Wl

where [0,t] is a large interval of R, i.e. t >> 0. Let T : X ^ X be the integral operator defined by

Vx e X Tx(t) = / g(|s - t|)x(s)ds, t e [0, t],

0

where g : (0, t] ^ R is a weakly singular function in the following sense:

(H) the function g is continuous and decreasing on (0,t], summable on [0,t], g(s) > 0 for all s e (0,t], lims^0+ g(s) =

Then T : X ^ X is a bounded operator [9], and the norm of T is given by

pr rT/2

llTIIbl(x) = sup llTxHx = 0maxT / g(|s - t|)ds = 2/ g(s)ds,

NxNx = 1 — — J 0 J 0

where BL (X) is the Banach space of all bounded operators from X to itself. Let A G C* be such that

r t/2

|A| = 2ß / g(s)ds, ß > 1, Jo

then A is in the resolvent set of T .By Neumann's theorem, we obtain that (A I — T )-1 exists, and

1

IKAI — T)-1 IUX) <

BL(X>" |A|-||T||bl(X)'

where I is the identity operator on X. Then the integral equation

Au(t) = f g(|s - t|)u(s)ds + f (t), t G [0, t], (0.2)

Jo

has a unique solution u G X for every f G X. Equation (0.2) is of great interest to mathematicians [9-11]. Our goal is to research this equation.

The paper is organized as follows. In Section 2, we introduce some notation and preliminary results. In Section 3, by using the previous results, we show how to formulate system (0.1). In Section 4, we treat our method of generalization of the Gauss-Seidel method in collocation with the product integration method. Finally, we give numerical results developed and compare our method with the conventional Gauss-Seidel method.

1. Notions and preliminary results

For N > 2 , we define a subdivision of the interval [0, t] by:

H = N tj = (j - 1) H, 1 < j < N + 1.

Let | ^Xj, IHijj j , N > 2 , be a family of Banach spaces, where Xj = C ([tj,tj+i]) is associated with the following norm:

Vx G X,- llxll. = max |x(t)| .

^ j j <i<ij+i

For 1 < i,j < N, we specify the Banach space B j = BL (Xj, Xj) of all bounded operators from Xj to Xj with the operator norm

VS G Bjj IISiijj = sup ||Sx|j.

IMI .=1

N

Let XN = JJ^ Xj be the product Banach space equipped with the norm j=i

VZ = (zi, ..., zN) G x^n ||Z ¡XX^ = 1mja?N ||zj .

Let BN = B^Xi^ be the Banach space of all bounded operators from XN to itself associated with the operator norm

VS G BN ||SII = sup |Sx|xxn . INI v =1

" IIXN

Let {Tj}1—ij<N be a family of operators defined by:

r tj+1

Tij : C([tj,tj+1]) ^ C([ti,ti+1]), x ^ Tijx(t) = g(|s - t|)x(s)ds, t E [ti,ti+1].

Jtj

It is clear that Tj e Bij- for all 1 < i, j < N, and

r j+1

11Tij||ij = max / g(|s - t|)ds.

j j tefti.ti+iJtj

For all 1 < i < N, ||Tii|ii < |A| , then (A/ii — Tii) is a bijection on Xi, its inverse is a bounded linear operator (see [11]), and

11 ||(A1ii — Tii) 11 ii <

|A| — ¡Ti^ii '

where Iii is the identity operator on Xi.

Lemma 1.1. For all 1 < i,j < N, we have:

f H/2

2 g(s)dS f i = j 0

r tj+1-ti+1

||Tij llij = / g(s)dS f i < j

tj —ti+1

Cti — tj

/ g(s)ds f i >j. „ 'ti — tj + 1

Proof. Consider the following function G : [0, t] ^ R that will play an important role in the proof:

G(t) := [ g(s)ds.

0

Case 1: i=j. Let y(t) : [ti,ti+1] ^ R be defined by

pti+1 rt fti+1

y(t) := / g(|s - t|)ds = g(t - s)ds + / g(s - t)ds = G(t - ti) + G(ti+1 - t).

ti ti t

The function y(t) is symmetric with respect to ti+^i+1 , and

y'(t) = g(t - ti) - g(ti+1 - t). Obviously, y'(t) > 0 if ti < t < , and y'(t) < 0 if ti+J*1 < t < ti+1. Hence

||Tii|ii = max y(t) = ti +ti+1) = 2 f g(s)ds.

ti<t<ti+1 2 0

Case 2: i<j. Let y(t) : [ti,ti+1] ^ R be defined by

r tj+i

y(t) := / g(s - t)ds = G(tj+1 - t) - G(tj - t).

tj

Then

y'(t) = g(tj - t) - g(tj+i - t) > 0 for all tj <t<tj+i.

Hence

rtj + 1-ti+1

||Tjj11 jj = max y(t) = y(tj+i) = g(s)ds.

ii<i<ii+1 Jtj-ti+1

Case 3: i>j. Let y(t) : [tj,tj+i] —> R be defined by

r tj+i

y(t) := / g(t - s)ds = G(t - tj) - G(t - tj+i). Jtj

Then

y'(t) = g(t - tj) - g(t - tj+i) < 0 for all tj < t < tj+i.

Hence

i'ti-tj

||Tjj i|jj = max y(t) = y(tj) = / g(s)ds.

ii<i<ii+i Jt-—t

j+i

□

Theorem 1.1. For integers N > 2, 1 < i < N, consider the positive parameters y., y., ,5H(i,N) and ,5* given by:

EIITy II« EIIT

ij IIj

_ j<i _ _ j>i Yi = |A| - IITiiHii' Yi = |A| - IITiiHii;

Y •

ßn(i, N) = y + Yi, ß, = max a

i<j<N 1 — y -i-j

We have

(i, N) < 1 and & < 1. Proof. Using the formulae obtained in Lemma 1.1, we get

N j-i N

E ||Tjj ¡jj E ||Tjj ¡jj + E ||Tjj ¡jj

(i N) = j=i,j=j_= j=i_j=j+i_

|A| - 11 Tjj1 jj |A| - 11 Tjj^jj

j-i rti-tj N ftj+i-ti+i

E / g(s)ds + E / g(s)ds

_ j=i «/ ti-tj+i j=j+i J tj-ti+1

|A| - 11 Tjj 1 jj

¡•ti rr-ti+i

/ g(s)ds + / g(s)ds

yH (ti)

/•r/2 /-H/2 /.r/2 /-H/2

2^ / g(s)ds - 2 g(s)ds 2^ / g(s)ds - 2 g(s)ds

Jo ,/0 ,/0 ,/0

where

¡•ti ¡-r-ti+1

yH(tj) := / g(s)ds + / g(s)ds = G(tj) + G(t - tj+i), 1 < i < N. Jo Jo

The sequence yH (tj) is symmetric with respect to 2 or 2 - H, and

yH(tj) = g(tj) - g(T - tj+i).

0

It is obvious, that yH(t) > 0 if 0 < t < 2 — H, and yH(t) < 0 if 2 < t < t. Hence

T T Z*t/2 rT/2 -H

i<i<N (ti) = y^2 — H = (^ = yo g(s)ds + Jo

g(s)ds.

As N > 2 then H < 2 , so we obtain

max yH (tj)

1<<N

^H (i,N) < - T/2 , H/2 '

2^ / g(s)ds — 2 g(s)ds ,/0 ,/0 T/2 T/2 -H

J g(s)ds + I g(s)ds

/•t/2 /.H/2

2^ / g(s)ds — 2 g(s)ds 00

< I < i

Finally, since (i, N) < 1, it is clear that ,5* < 1.

□

2. Formulation of system (0.1)

In this section, we see how to formulate system (0.1) according to our integral equation. Let {uj }1<j<N be a family of continuous functions such that

u

j e X, Vt e [tj ,tj+i] Uj (t) = u(t).

We have

Au(t)= f g(|s — t|)u(s)ds + /(t) = ^ /J+1 g(|s — t|)Uj (s)ds + /(t), t e [0, t] Jo j=i Jtj

which is equivalent to the same system (0.1) described in the introduction:

f Aui(t) = Tiiui(t)+ Ti2u2(t) + ... + Tinun(t) + fi(t), t e [ti,t2], Au2(t) = T2iui(t) + T22u2(t) + ... + T2NuW(t) + /2<t), t e [t2,ts],

k Aun (t) = Tniui(t) + Tn2u2(t) + ... + Tnnun(t) + /w (t), t e [tw, tw +i], where

/j e Xj, Vt e [tj,ti+i] /j(t) = /(t).

This system is equivalent to the following linear equation:

AU = MT U + F, where MT : Xw ^ Xw is the operator matrix defined by

MT =

/Tn Ti2 T2i T22

Tin \

T2N

\TN i TN 2 ... TNN /

F = (/i,..., /N) is given in XN and U = («i,..., uN) is to be found in the same space. It is clear that MT is a bounded operator. We have

N N rj+i

||MT|| = max > ||T„||.. = max > max / g(|s — t|)ds

" T 11 i<i<Nj=i " jllj i<i<Nj=ite[ti>ti+i]Jt. yv|

maW g(|s — t|)ds < |A| .

0<t<T J0

We use Neumann's theorem [12] to conclude that (A/N — MT) 1 exists and

|(A/n - MT)-1|| <

|A| — || Mt II'

where 1N is the identity operator on XN. This assures the existence and uniqueness of the solution U =(u1,...,uN) of the system (0.1) for all F =(/1,...,/N) in XN .

3. Generalized Gauss—Seidel method

In this section, we construct a generalization of the Gauss-Seidel method suitable for our system (0.1).

3.1. Definition of an iterative sequence and its convergence

Consider the following iterative scheme:

AUk = LtUk + (MT — LT) Uk-1 + F, k > 1,

U0 G Xn

where is the lower triangular matrix part of MT defined by

L

T

(Tii Oi2 ... OiN \

T21 T22 ... O2N

\TN 1 TN 2 ... TNN /

For 1 < i, j < N, Oj : Xj ^ Xj is the null operator, i.e. Vx G Xj, Ojx = 0Xi. We can write the precedent iterative scheme in a simple and clear formula: for 1 < i < N,

u0 e X

with

i-1 N

Auk(t) = Tiiuk(t) + ETijuk(t) + E Tijuk-1(t) + fi(t), t e Mi+i], k > 1, j=1 j=i+1

0

0N

ETij uk = E Tij uk = 0xi.

j=1 j=N+1

Our goal is to prove that Uk ^ U for k ^ œ.

Theorem 3.1. We have

lim^ || Uk — U||xN = 0.

1

P r o o f. For all 1 < i < N ,

j-i N

Auj = TjjUj + ^ TjjUj + ^ TjjUj j=i j=j+i j- i N

(A/jj - Tjj)uj = Tjj Uj + Tjj j=i j=j+i

j-i N

Uj = (A/jj - Tjj)-i ^ Tjj Uj + (A/jj - Tjj)-i ^ Tjj Uj + (A/jj - Tjj)-ifj. j=i j=j+i

In the same way, we get

j- i N

uk = (A/jj - Tjj)-i ^ TjjUk + (A/jj - Tjj)-i ^ Tjjuj-1 + (A/jj - Tjj)-ifj. j=i j=j+i

Then

j- i N

(uk - Uj) = (A/jj - Tjj)-i ^ Tjj(u£ - Uj) + (A/jj - Tjj)-i ^ Tjj(uj-1 - Uj).

j=i j=j+i

Therefore

j-i N

|uk - ujnj < ||(A/jj- Tjj)-1|j^ |Tjj¡jj¡ujfc - u,¡j + ¡(A/jj-Tjj)-1|jj Y^ |Tjj¡jj¡uj-1 -u,¡j

j=1 j=j+1 j-1 N

E 1 Tjj 11 jj E 1 Tjj 1 jj

j=i k j=j+i k i

" |A| - 11Tjj|jj ¡Uj - Uj ||j + |A| - |Tjj||jj ¡Uj - Uj||j < Yj|Uk - U¡Xxn + Yj|Uk-i - U¡XXn. Let im G N be such that

¡Ujm - Ujm ¡Xim = |U' - U ¡XXN .

We obtain

¡Uk - U¡Xxn < 7jm |Uk - U¡XXn + Yjm |Uk-i - U|Xn, (1 - Y jm )|Uj - U^Xn < Y jm |Uj-i - U ¡Xn ,

|Uj - U¡XXn < Y^|Uj-i - U¡Xn,

—jm

|Uj - U¡XXn < ft||Uj-i - U¡XXn. Repeating this operation k times, we find that

|Uk - UiXn < tf |U0 - UiXXn. Now we use the fact that ,5* < 1 to conclude the proof. □

To get an approximation of the solution u, we construct it, for the k-th iteration, using the following formula:

Vt G [tj, tj+i) u(t) « uj(t), 1 < i < N,

and u(b) = uN(b).

3.2. Product integration method

In practice, for 1 < i < N, (A/^ — T^)-1 can not be found exactly. For that, we need to approximate it using product integration method [10]. It will be easy, because, for 1 < i < N, [ii,ii+1] is not very large compared to [0,t] .

First of all, we study the following equation in order to explain the product integration method: for 1 < i < N

Vt e [tj, ii+i] Auj(t) = TjiUi(i) + gi(t).

It is clear that u e Xj exists and is unique for all g e Xj. For n > 2 , 1 < i < N, we define a subdivision of [ti,ti+1 ] by

H

hn =-7, Sj,p = tj + (p — 1)hn, 1 < p < n.

n — 1

For 1 < i < N, let |ei,p(s)}n=1 C Xj be such that for 2 < p < n — 1,

-i _ |s Si,p| s < ^ < s

ei,p(s) = { 1 h„ ' Si,p-1 " S " Si,p+1

0, otherwise.

Sj 2 — S

/ \ I —~L-, Sj,1 < S < Sj,2

£¿,1 (s) = < h„

0, otherwise.

s Sj,n—1 ^ ^

/ \ i T , sj,n—1 < s < sj,n

ej,n(s) ^ hn

0, otherwise. Let Tjj,n : Xj ^ Xj be a linear operator defined by

n

Vx e Xj Tjj,nx(t) = ^ wj,p(t)x(sj,p), t e [tj, tj+1 ], p=1

with weights

f ii+1

Wj,p(t) = / g(|s — t|)ej,p(s)ds.

Jti

From [10], it follows that for 1 < i < N, Tjj,n e Bjj and

f H/2

llT^nlla < 2 g(s)ds. Jo

Let us denote

r H/2

¿h := 2 g(s)ds.

o

Theorem 3.1. For all 1 < i < N, for n large enough, (A/jj — Tjj,n)—1 exists and

||(A/jj — Tjj,n) <

where k is a positive constant independent of i and n.

Proof. For all 1 < i < N, using Lemma 4.1.2 [10], we get

lim ||(Tjj - Tjj,ra)Tjj,ra|| = 0.

From Theorem 4.1.2 [10], we obtain that (A/.. — Tii,n)-1 exists for n large enough and II (A/ii Tü,ra) 11.-.- <

,-1M /1 + ||(A1ii — Tii)-1||ii ||T«

But

Then

||T«,„|„ < Sh, ||(A1 — Tii)-1||ii <

|A| — ||(Tii — Tii,n)Tii 1

IA | — ||Tii||ii | A | — Sh

lim 1 + ||(A/ii Tii) ||Tii,n||ii < |a|—1 (1 + (|a| — )-1 5h)

n^ | A | — ||(Tii — Tjj,ra)Tjj,ra|ii < v 1

To conclude the proof, we take

k =1 + |A| 1 (1 + (|A| — ¿h)—1 6H) .

□

For 1 < i < N, for n large enough, let uj)n G Xj be a unique solution of the following equation:

Aui,n(t) = Tji)nMi,n(t) + gi(t), t G [tj, tj+i]. Theorem 3.2. For 1 < i < N, for n large enough,

||«j — Uj,n|j < ^HKW (hn,«j) ,

where w (hn,uj) is the modulus of continuity of uj(t) on [tj,tj+1] defined by

w (hn,uj)= max |uj(t) — uj(s)|.

|s-t|<hn

Proof. Follows from Theorem 4.2.1 in [10] and the fact that

«j uj,n (A1jj Tjj,ra) (Tjj Tjj,ra) «j.

For 1 < i < N, uj)n is calculated by the formula

1

□

Vt e [ti,ti+1] ui,n(t) = a(Ewi,p(t)xp + gi(t))

p=1

where x = (x1, ...,xp) is a unique solution of the system

Ax = Ax + b, Aqp = Wi>p(si>g), bq = gi(si;q), 1 < < n.

1

We define the iterative scheme of the product integration version of the generalized Gauss-Seidel method corresponding to our integral equation by

i-1 N

Kn(t) = Tkn^Jt) + YTijUjn(t) + Y Tjuk-1(t) + fi(t), t e [ti,ti+i], k > 1,

j=1 j=i+1

u° e Xi,

for 1 < i < N and n > 2. For k > 0, we define Uk = (uk,n, u2,n,..., u^n) e XN. For technic n > 2,

technical reasons, we need to define U^ = (ukn,U2n,... ,uN n) e XN : let for 1 < i < N and

i- 1 N

AU|,Jt) = Tiiû|,n(t) + YTijuk,n(t) + Y Tij«j- + fi(t), t e [ti,ti+1 ], k > 1,

j=1 j=i+1

u° e Xi.

Theorem 3.3. For k > 1 , n > 2 , we have

llUj —UIIxn < max w(hn,ûi,J + ßj||U° —U||^ ,

^ °< 2<j

where ê :— I I

|A| — ||Mt|| '

Proof. We have, for n > 2 ,

llUj — UIIXXn < lU — UjIIXXN + llUj — UIIXXN.

But, for 1 < i < N,

i- 1 N

(ujn — Ui) = (A/« — T«)-1 Y Tij (j — Uj) + (A/« — T«)-1 Y Tij (uj- — Uj).

j=1 j=i+1

Therefore

I «j,n «i H i < ||(A/ii Tii) 11 ii ^ ] ||Tij 11 ij 11 « j,n Uj

% Hi? ||uj,n Uj 11 j

Let im G N be such that

i-1

< H(A/ü — Tii)-1Hii y iit

j=1

N

+ 11 (A/ii — Tii)-1Hii Y l|Ti

j=i+1

< YiHUnj — UiixXn + YiHUj-1 — UiiXXn.

I«jm,n — HXim = 11 U„ — U H ^ .

We obtain

llU2 — u< Yj llU2 — U||*v + Yjm llU2—1 — Ull

Then

IlUk - U iixTn < IlUk - iiXXn + Yim l^ - U hXTn + Yim HUi-1 - U iiXn , iiu?-uiix^N < y-^iiuk-ukiixN + IIuk-1 -Uii^

—im —im

< tffek max " (h„,«y + ß, ||Uk-1 - U||jN.

Repeating the last inequality,

IlUk - UIIxn < kmaxw(h„,uk;„) + ß*(^HK max "(hn,«?-1) + ß,IIU?-2 - UH^

fc-i

< tffekEß, 1m£aXv" (hn,^) + ß?IIU0 - UIIxn

1=0

< T^Hr max" (hn,«i,n)+ß?iiu0 - uiixn. i - ß i<i<N ^ ^ 0< Kfc

□

4. Numerical Results

We illustrate the application of our numerical method by considering the following Fredholm integral equation of the second kind:

r 100 . (s)

A«(t)=/ . ( ) ds + f (t), A = 4^v/2, t G [0,100], Jo Vis - t|

where _

.(t) = At2 2(3 x 104 + 400t + 8t2)Vl00 - t 1615/2

f ( ) = 15 15 ,

and the exact solution is u(t) = t2 on [0,100]. The kernel g(s) = ^ satisfies the hypothesis

(H). We mention that this equation is the same studied in [8].

In order to give a comprehensive view of the procedure of the generalized versions, we study

this example by applying the following methods:

1. Generalized Gauss-Seidel method, our method described in this paper.

2. Conventional Gauss-Seidel method, the method described in [2], the latter applies the Gauss-Seidel iterative scheme to approach the huge matrix obtained after using the product integration method.

3. Generalized Jacobi method, the method described in [8].

We fix H = 1 and we take the null function as a starting point for our method and the null vector for the conventional Gauss-Seidel method. The stopping condition on the parameter k is fixed by

||Unew - UwH < 10-8.

Eggs'(hn) , (hn) and EGJ(hn) denote the absolute error obtained by using the Generali-

zed Gauss-Seidel method, Conventional Gauss-Seidel method and Generalized Jacobi method, respectively. We vary now to compare the results of the methods.

Table 1. Numerical results

hn egg,S

0.250 9.86E-3 1.20E-3

0.125 2.47E-3 3.12E-4

0.050 3.97E-4 5.15E-5

0.025 1.01E-4 1.38E-5

Table 2. Numerical results

hn EGj (Results of [8]) k EGGS' k

0.250 1.20E-3 34 1.20E-3 20

0.125 3.12E-4 34 3.12E-4 20

0.050 5.15E-5 34 5.15E-5 20

0.025 1.38E-5 34 1.38E-5 20

Table (1) shows that the error committed by the two methods decreases with the decrease of hn, but the error order of the generalized version of Gauss-Seidel method is smaller than the error order of the conventional version of Gauss-Seidel method. Furthermore, in Table (2) we can also see that the both generalized methods (Gauss-Seidel and Jacobi) give the same results, but the generalized Gauss-Seidel method is faster than the generalized Jacobi method. So, we confirm that our vision of generalization is reasonable.

5. Concluding remarks

We have constructed a generalization of the Gauss-Seidel iterative method for a system of linear operators. We used this new technique, in collocation with the product integration method, to approximate a solution of the Fredholm linear integral equation of the second kind with a weakly singular kernel on a large interval. The numerical tests show the efficiency of our new method compared to the classical Gauss-Seidel method.

References

[1] W. Li, W. Sun, "Modified Gauss-Seidel type methods and Jacobi type methods for Z-matrices", Linear Algebra and Its Applications, 317:1 (2000), 227-240.

[2] M. S. Muthuvalu, "The preconditioned Gauss-Seidel iterative methods for solving Fredholm integral equations of the second kind", AIP Conference Proceedings, 1751 (2016), 020001.

[3] Y. Saad, Iterative Methods for Sparse Linear Systems, 2-nd ed., Society for Industrial and Applied Mathematics, Siam, 2003, 567 pp.

[4] D.K. Salkuyeh, "Generalized Jacobi and Gauss-Seidel methods for solving linear system of equations", Numer. Math. J. Chinese Univ., 16:2 (2007), 164-170

[5] Y. Zhang, T. Z. Huang, X. P. Liu, "Modified iterative methods for nonnegative matrices and M-matrices linear systems", Computers & Mathematics with Applications, 50:10 (2005), 1587-1602

[6] L. Zou, Y. Jiang, "Convergence of The Gauss-Seidel Iterative Method", Procedia Engineering, 15 (2011), 1647-1650

[7] S. Lemita, H. Guebbai, "New process to approach linear Fredholm integral equations defined on large interval", Asian Eur. J. Math., 12:01 (2019), 1950009

[8] S. Lemita, H. Guebbai, M. Z. Aissaoui, "Generalized Jacobi method for linear bounded operators system", Comput. Appl. Math., 37:3 (2018), 3967-3980

[9] M. Ahues, A. Largillier, O. Titaud, "The roles of a weak singularity and the grid uniformity in relative error bounds", Numerical Functional Analysis and Optimization, 22 (2001), 789-814

[10] K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, United Kingdom, 1997

[11] K. Atkinson, W. Han, Theoretical Numerical Analysis: A Functional Analysis Approach, Springer, New York, 2009

[12] M. Ahues, A. Largillier, B. V. Limaye, Spectral Computations for Bounded Operators, Chapman and Hall/CRC, New York, 2001

Information about the authors

Samir Lemita, PhD, Assistant Professor. Higher Normal School of Ouargla, Ouargla, Algeria. E-mail: [email protected] ORCID: https://orcid.org/0000-0003-2568-2493

Hamza Guebbai, Full Professor. University May 8, 1945 - Guelma, Guelma, Algeria. E-mail: [email protected]

ORCID: https://orcid.org/0000-0001-8119-2881

Ilyes Sedka, Post-Graduate Student. University May 8, 1945 - Guelma, Guelma, Algeria. E-mail: [email protected]

Mohamed Zine Aissaoui, Full Professor. University May 8, 1945 - Guelma, Guelma, Algeria. E-mail: [email protected] ORCID: https://orcid.org/0000-0001-5253-9671

There is no conflict of interests.

Corresponding author:

Samir Lemita

E-mail : [email protected]

Received 18.07.2020 Reviewed 15.09.2020 Accepted for press 19.11.2020

Информация об авторах

Лемита Самир, PhD, доцент. Высшая нормальная школа Уаргла, г. Уаргла, Алжир. E-mail: [email protected]

ORCID: https://orcid.org/0000-0003-2568-2493

Геббай Хамза, профессор. Университет 8 мая 1945 г. - Гельма, г. Гельма, Алжир. E-mail:[email protected] ORCID: https://orcid.org/0000-0001-8119-2881

Седка Ильес, аспирант. Университет 8 мая 1945 г. - Гельма, г. Гельма, Алжир. E-mail:[email protected]

Аиссауи Мохамед Зин, профессор. Университет 8 мая 1945 г. - Гельма, г. Гельма, Алжир. E-mail: [email protected] ORCID: https://orcid.org/0000-0001-5253-9671

Конфликт интересов отсутствует.

Для контактов:

Лемита Самир

E-mail: [email protected]

Поступила в редакцию 18.07.2020 г. Поступила после рецензирования 15.09.2020 г. Принята к публикации 19.11.2020 г.